You might also be able to rewrite log(1 - cos(x))
as log1p(-cos(x)) On Wed, Aug 16, 2023, 02:51 Iris Simmons <ikwsi...@gmail.com> wrote: > You could rewrite > > 1 - cos(x) > > as > > 2 * sin(x/2)^2 > > and that might give you more precision? > > On Wed, Aug 16, 2023, 01:50 Leonard Mada via R-help <r-help@r-project.org> > wrote: > >> Dear R-Users, >> >> I tried to compute the following limit: >> x = 1E-3; >> (-log(1 - cos(x)) - 1/(cos(x)-1)) / 2 - 1/(x^2) + log(x) >> # 0.4299226 >> log(2)/2 + 1/12 >> # 0.4299069 >> >> However, the result diverges as x decreases: >> x = 1E-4 >> (-log(1 - cos(x)) - 1/(cos(x)-1)) / 2 - 1/(x^2) + log(x) >> # 0.9543207 >> # correct: 0.4299069 >> >> I expected the precision to remain good with x = 1E-4 or x = 1E-5. >> >> This part blows up - probably some significant loss of precision of >> cos(x) when x -> 0: >> 1/(cos(x) - 1) - 2/x^2 >> >> Maybe there is some room for improvement. >> >> Sincerely, >> >> Leonard >> ========== >> The limit was part of the integral: >> up = pi/5; >> integrate(function(x) 1 / sin(x)^3 - 1/x^3 - 1/(2*x), 0, up) >> (log( (1 - cos(up)) / (1 + cos(up)) ) + >> + 1/(cos(up) - 1) + 1/(cos(up) + 1) + 2*log(2) - 1/3) / 4 + >> + (1/(2*up^2) - log(up)/2); >> >> # see: >> >> https://github.com/discoleo/R/blob/master/Math/Integrals.Trig.Fractions.Poly.R >> >> ______________________________________________ >> R-help@r-project.org mailing list -- To UNSUBSCRIBE and more, see >> https://stat.ethz.ch/mailman/listinfo/r-help >> PLEASE do read the posting guide >> http://www.R-project.org/posting-guide.html >> and provide commented, minimal, self-contained, reproducible code. >> > [[alternative HTML version deleted]] ______________________________________________ R-help@r-project.org mailing list -- To UNSUBSCRIBE and more, see https://stat.ethz.ch/mailman/listinfo/r-help PLEASE do read the posting guide http://www.R-project.org/posting-guide.html and provide commented, minimal, self-contained, reproducible code.
